Area of Circles and Sectors
We already know that area is the space inside a 2D shape. We can find the area of a circle, but we will need a special rule.
The following investigation will demonstrate what happens when we unravel segments of a circle.
Interesting isn't it that when we realign the segments we end up with a parallelogram shape. Which is great, because it means we know how to find the area based on our knowledge that the area of a parallelogram has formula $A=bh$A=bh. In a circle, the base is half the circumference and the height is the radius.
$\text{Area of a circle}=\pi r^2$Area of a circle=πr2
What if we don't have an entire circle?
Well, half a circle would have half the area or half the circumference. One quarter of a circle would have a quarter of the area, or a quarter of the circumference. In fact all we need to know is what fraction the sector is of a whole circle. For this all we need to know is the angle of the sector.
Looking at the quarter circle, the angle of the sector is $90$90°. The fraction of the circle is $\frac{90}{360}=\frac{1}{4}$90360=14.
More generally, If the angle of the sector is $\theta$θ, then the fraction of the circle is represented by
$fraction=\frac{\theta}{360}$fraction=θ360 (due to there being $360$360° in a circle).
Example
Question: Find the area of a sector with central angle of $126$126° and radius of $7$7cm. Evaluate to $2$2 decimal places.
Think: What fraction is this sector of a whole circle? What is the rule for area?
Do: This sector is $\frac{126}{360}=0.35$126360=0.35 of a circle.
Area of a circle is $A=\pi r^2$A=πr2, so the area of the sector is
| $0.35\times\pi r^2$0.35×πr2 | $=$= | $0.35\pi\times7^2$0.35π×72 |
| $=$= | $17.15\pi$17.15π | |
| $=$= | $53.88$53.88 cm2 (rounded to 2 decimal places) |
More Worked Examples
Question 1
If the radius of the circle is $5$5 cm, find its area.
Give your answer as an exact value.
Question 2
Find the area of the shaded region in the following figure, correct to one decimal place.
Question 3
Find the area of the shaded region in the following figure, correct to one decimal place.
Question 4
Consider the sector below.
Calculate the perimeter. Give your answer correct to two decimal places.
Calculate the area. Give your answer correct to two decimal places.
Question 5
Consider the sector below.
Calculate the perimeter. Round your answer to two decimal places.
Calculate the area. Round your answer to two decimal places.
QUESTION 6
A goat is tethered to a corner of a fenced field (shown). The rope is $9$9 m long. What area of the field can the goat graze over?
Give your answer correct to two decimal places.